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This is a book about a special kind of geometry that was invented and widely practiced in Japan during the centuries when Japan was isolated from Western infl uences. Japa nese geometry is a mixture of art and mathematics. The experts communicated with one another by means of sangaku, which are wooden tablets painted with geometrical fi gures and displayed in Shinto shrines and Buddhist temples. Each tablet states a theorem or a problem. It is a challenge to other experts to prove the theorem or to solve the problem. It is a work of art as well as a mathematical statement.
Sangaku are perishable, and the majority of them have decayed and disappeared during the last two centuries, but enough of them have survived to fi ll a book with examples of this unique Japa nese blend of exact science and exquisite artistry. Each chapter of the book is full of interesting details, but for me the most novel and illuminating chapters are 1 and 7. Chapter 1 describes the historical development of sangaku, with emphasis on Japan’s “peculiar institution,” the samurai class who had originally been in de pen dent warriors but who settled down in the seventeenth century to become a local aristocracy of well- educated offi cials and administrators.

It was the samurai class that supplied mathematicians to create the sangaku and work out the problems. It is remarkable that sangaku are found in all parts of Japan, including remote places far away from cities. The reason for this is that samurai were spread out all over the country and maintained good communications even with remote regions. Samurai ran schools in which their children became literate and learned mathematics. Samurai combined the roles which in medieval Eu rope were played separately by monks and feudal lords. They were scholars and teachers as well as administrators.
Chapter 7 is my favorite chapter, the crown jewel of the book. It contains extracts from the travel diary of Yamaguchi Kanzan, a mathematician who made six long journeys through Japan between 1817 and 1828, recording details of the sangaku and their creators that he found on his travels. The diary has never been published, but the manuscript is preserved in the archives of the city of Agano. The manuscript runs to seven hundred pages, so that the brief extracts published here give us only a taste of it.
It is unique as a fi rst- hand eye- witness description of the sangaku world, written while that world was still at the height of its fl owering, long before the sudden irruption of Western culture and modernization that brought it to an end. I hope that the diary will one day be translated and published in full. Meanwhile, this book, and chapter 7 in par tic u lar, gives us a glimpse of Yamaguchi Kanzan as a mathematician and as a human being. Having been present at the creation, he brings the dead bones of sangaku to life. I am lucky to have known two scholars who have devoted their lives to cultivating and teaching geometry.

They are Daniel Pedoe in En gland and the United States, and Fukagawa Hidetoshi in Japan. Each of them had to swim against the tide of fashion. For the last fi fty years, both in art and mathematics, the fashionable style has been abstract: famous artists such as Jackson Pollock produce abstract patterns of paint on canvas; famous mathematicians such as Kurt Gödel construct abstract patterns of ideas detached from anything we can feel or touch. Geometry is like repre sen ta tional painting, concerned with concrete objects that have unique properties and exist in the real world. Fashionable artists despise repre sen ta tional painting, and fashionable mathematicians despise geometry. Repre sen ta tional painting and geometry are left for amateurs and eccentric enthusiasts to pursue.